This equation is a particular case of a depressed
quartic equation and it can be solved by the Ferrari method, hence reducing it to a depressed cubic equation, and then use Cardano's formulas.
The stability of the endemic equilibrium [E.sub.1] is difficult to prove analytically, because it involves a
quartic equation which depend on the variables [I.sub.m] and [I.sub.f].
Hence, from equation (1) it is soon seen that x satisfies the
quartic equationwhich is a
quartic equation in general [11] and its form depends on the medium bidyadic [??].
According to the solution of a
quartic equation in [21], we can obtain the following solutions:
Case 2: A
quartic equation with two real roots and one imaginary root of multiplicity 2
Now [X.sub.jkt] includes year fixed effects and a
quartic equation in free throw attempts.
The geometric solution of a
quartic equation by the use of circles and y = [x.sup.2] has been presented.
The
quartic equation A = 0 has roots a = [+ or -](1 + [square root of (1 - [[kappa].sup.2])]v/2[[kappa].sup.2] and [+ or -](1 - [square root of (1 - [[kappa].sup.2])]v/2[[kappa].sup.2] with [a.sub.2] = (2 - [[kappa].sup.2] + 2[square root of (1 - [[kappa].sup.2])]v/2[[kappa].sup.2]/4[[kappa].sup.4] and (2 - [[kappa].sup.2] - 2[square root of (1 - [[kappa].sup.2])][v.sup.2]/4[[kappa].sup.4], respectively.
We will use the Booker
quartic equation derived by Pettis [12] for [k.sub.z], given by